Extra properties

Extra structure

Contents

Idea

A graph is reflexive if for each vertexvv there is a (specified) edgev→vv \to v.

Properties

Proposition

The category of reflexive directed graphs RefGphRefGph, i.e., reflexive quivers, equipped with the functor U:RefGph→SetU: RefGph \to Set which sends a graph to its set of edges, is monadic over SetSet.

Proof

RefGphRefGph is the category of functors R→SetR \to Set where RR is the walkingreflexive fork, consisting of two objects 0,10, 1 and generated by arrows i:0→1i: 0 \to 1 and s,t:1→0s, t: 1 \to 0 subject to si=10=tis i = 1_0 = t i and no other relations. This RR is in turn the Cauchy completion of a monoid MM consisting of two elements e0=is,e1=ite_0 = i s, e_1 = i t and an identity, with multiplication e0e0=e0=e1e0e_0 e_0 = e_0 = e_1 e_0 and e1e1=e1=e0e1e_1 e_1 = e_1 = e_0 e_1, and therefore RefGphRefGph is equivalent to the category of functors M→SetM \to Set, i.e., the category of MM-sets SetMSet^M. This is the category of algebras of the monad M×−M \times - whose monad structure is induced from the monoid structure of MM.